Leonid I. Slepyan

The dynamic response of elastic waveguides is important for a wide range of applications that involve dispersive waves as well as wave localization. In particular, a case of special interest relates to waveguides subjected to moving loads. In the case where the elongated structure includes a sequence of built-in resonators, the range of resonance regimes may be extended accordingly. The present paper gives an overview of several mathematical formulations that connect Floquet theory to the dynamic response of multi-scale waveguides, which include inertial sub-structures subjected to external forces. This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)’.

This paper presents a unified approach to the modelling of elastic solids with embedded dynamic microstructures. General dependences are derived based on Green's kernel formulations. Specifically, we consider systems consisting of a master structure and continuously or discretely distributed oscillators. Several classes of connections between oscillators are studied. We examine how the microstructure affects the dispersion relations and determine the energy distribution between the master structure and microstructures, including the vibration shield phenomenon. Special attention is given to the comparative analysis of discrete and continuous distributions of the oscillators, and to the effects of non-locality and trapped vibrations. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.

In this brief article (written at the suggestion of Prof. Gennady Mishuris), I discuss some projects, the circumstances under which the associated problems appeared on my desk, the results and what helped me to obtain them. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.

We consider a segmented structure, possibly connected with a continuous medium, as initially homogeneous, where discontinuities arise as localized strains induced by self-equilibrated localized actions. Under this formulation augmented by interface conditions, the linearized formulation remains valid. This approach eliminates the need for examining separate sections with subsequent conjugation. Only conditions related to the discontinuities should be satisfied, while the continuity in other respects preserves itself automatically. No obstacle remains for the continuous Fourier transform. For a uniform partitioning, the discrete transform is used together with the continuous one. We demonstrate the technique by obtaining the Floquet wave dispersive relations with their dependence upon interface stiffness. To this end, we briefly consider the flexural wave in the segmented beam on Winkler's foundation, the gravity wave in a plate (also segmented) on deep water and the Floquet-Rayleigh wave in such a plate on an elastic half-space. Besides, we present the wave equations developed for an elastic medium with discontinuities.

We consider parallel splitting of a strip composed of two different chains. As a waveguide, the dissimilar-chain structure radically differs from the well-studied identical-chain system. It is characterized by three speeds of the long waves, two for the separate chains, and one for the intact strip where the chains are connected. Accordingly, there exist three ranges, the subsonic for both chains, the intersonic and the supersonic. The speed in the latter range is supersonic for the intact strip and at the same time, it is subsonic for the separate higher-speed chain. This fact allows the splitting wave to propagate in the strip supersonically. We derive steady-state analytical solutions and find that the splitting can propagate steadily only in two of these speed ranges, the subsonic and the supersonic, whereas the intersonic regime is forbidden. In the case of considerable difference in the chain stiffness, the lowest dynamic threshold corresponds to the supersonic regime. The peculiarity of the supersonic mode is that the supersonic energy delivery channel, being initially absent, is opening with the moving splitting point. Based on the discrete and related continuous models we find which regime can be implemented depending on the structure parameters and loading conditions. The analysis allows us to represent the characteristics of such processes and to demonstrate strengths and weaknesses of different formulations, quasi-static, dynamic, discrete or continuous. Analytical solutions for steady-state regimes are obtained and analyzed in detail. We find the force-speed relations and show the difference between the static and dynamic thresholds. The parameters and energy of waves radiated by the propagating splitting are determined. We calculate the strain distribution ahead of the transition point and check whether the steady-state solutions are admissible.